ipca: Independent Principal Component Analysis
In mixOmics: Omics Data Integration Project

Description Usage Arguments Details Value Author(s) References See Also Examples

Performs independent principal component analysis on the given data matrix, a combination of Principal Component Analysis and Independent Component Analysis.

ipca(
  X,
  ncomp = 2,
  mode = "deflation",
  fun = "logcosh",
  scale = FALSE,
  w.init = NULL,
  max.iter = 200,
  tol = 1e-04
)

`X`	a numeric matrix (or data frame).
`ncomp`	integer, number of independent component to choose. Set by default to 3.
`mode`	character string. What type of algorithm to use when estimating the unmixing matrix, choose one of `"deflation"`, `"parallel"`. Default set to `deflation`.
`fun`	the function used in approximation to neg-entropy in the FastICA algorithm. Default set to `logcosh`, see details of FastICA.
`scale`	(Default=FALSE) Logical indicating whether the variables should be scaled to have unit variance before the analysis takes place. The default is `FALSE` for consistency with `prcomp` function, but in general scaling is advisable. Alternatively, a vector of length equal the number of columns of `X` can be supplied. The value is passed to `scale`.
`w.init`	initial un-mixing matrix (unlike fastICA, this matrix is fixed here).
`max.iter`	integer, the maximum number of iterations.
`tol`	a positive scalar giving the tolerance at which the un-mixing matrix is considered to have converged, see fastICA package.

In PCA, the loading vectors indicate the importance of the variables in the principal components. In large biological data sets, the loading vectors should only assign large weights to important variables (genes, metabolites ...). That means the distribution of any loading vector should be super-Gaussian: most of the weights are very close to zero while only a few have large (absolute) values.

However, due to the existence of noise, the distribution of any loading vector is distorted and tends toward a Gaussian distribtion according to the Central Limit Theroem. By maximizing the non-Gaussianity of the loading vectors using FastICA, we obtain more noiseless loading vectors. We then project the original data matrix on these noiseless loading vectors, to obtain independent principal components, which should be also more noiseless and be able to better cluster the samples according to the biological treatment (note, IPCA is an unsupervised approach).

Algorithm 1. The original data matrix is centered.

2. PCA is used to reduce dimension and generate the loading vectors.

3. ICA (FastICA) is implemented on the loading vectors to generate independent loading vectors.

4. The centered data matrix is projected on the independent loading vectors to obtain the independent principal components.

ipca returns a list with class "ipca" containing the following components:

`ncomp`	the number of independent principal components used.
`unmixing`	the unmixing matrix of size (ncomp x ncomp)
`mixing`	the mixing matrix of size (ncomp x ncomp)
`X`	the centered data matrix
`x`	the independent principal components
`loadings`	the independent loading vectors
`kurtosis`	the kurtosis measure of the independent loading vectors
`prop_expl_var`	Proportion of the explained variance of derived components, after setting possible missing values to zero.

Fangzhou Yao, Jeff Coquery, Kim-Anh Lê Cao, Florian Rohart, Al J Abadi

Yao, F., Coquery, J. and Lê Cao, K.-A. (2011) Principal component analysis with independent loadings: a combination of PCA and ICA. (in preparation)

A. Hyvarinen and E. Oja (2000) Independent Component Analysis: Algorithms and Applications, Neural Networks, 13(4-5):411-430

J L Marchini, C Heaton and B D Ripley (2010). fastICA: FastICA Algorithms to perform ICA and Projection Pursuit. R package version 1.1-13.

sipca, pca, plotIndiv, plotVar, and http://www.mixOmics.org for more details.

data(liver.toxicity)

# implement IPCA on a microarray dataset
ipca.res <- ipca(liver.toxicity$gene, ncomp = 3, mode="deflation")
ipca.res

# samples representation
plotIndiv(ipca.res, ind.names = as.character(liver.toxicity$treatment[, 4]),
group = as.numeric(as.factor(liver.toxicity$treatment[, 4])))

## Not run: 
plotIndiv(ipca.res, cex = 0.01,
col = as.numeric(as.factor(liver.toxicity$treatment[, 4])),style="3d")

## End(Not run)
# variables representation
plotVar(ipca.res, cex = 0.5)

## Not run: 
plotVar(ipca.res, rad.in = 0.5, cex = 0.5,style="3d")

## End(Not run)